Sparse grid methods applied to solve partial diierential equations allow for a substantial reduction of numerical eeort (to obtain equal error magnitudes) compared to conventional nite element methods. The stationary Schrr odinger equation is solved numerically for a number of model problems and the results are compared to both exact values and numerical computations of other authors. For probl...

We consider an optimal control problem with indeenite cost for an abstract model which covers in particular hyperbolic and hyperbolic-like systems in a general bounded domain. Necessary and suucient conditions are given for the synthesis of the optimal control, which is given in terms of the Riccati operator arising from a nonstandard Riccati equation. The theory also extends a nite-dimensional...

Based on a q-deformation of the kinematical algebra C 1 (S 1) +Vect(S 1) a q-analog to the Borel quantization on S 1 is constructed. The resulting momentum operator is given in terms of q-derivatives. Via a generalized version of the Ehrenfest theorem it leads, together with the position operator, to a nonlinear q-diierence equation on the N-point discretization S 1 N of S 1. In the limit q ! 1...

Schrr odinger equations with nonlinearities concentrated in some regions of space are good models of various physical situations and have interesting mathematical properties. We show that in the semiclassical limit it is possible to separate the relevant degrees of freedom by noticing that in the regions where the nonlinearities are eeective all states are suppressed but the metastable ones (re...

We develop the theory of hybrid fractional differential equations with the complex order $thetain mathbb{C}$, $theta=m+ialpha$, $0<mleq 1$, $alphain mathbb{R}$, in Caputo sense. Using Dhage's type fixed point theorem for the product of abstract nonlinear operators in Banach algebra; one of the operators is $mathfrak{D}$- Lipschitzian and the other one is completely continuous, we prove the exis...

A large number of di erential equation problems which admit traveling waves have very large (typically in nite) naturally de ned domains, with boundary conditions de ned at the domain boundary. To be able to numerically solve these problems in smaller subdomains of the original domain, arti cial boundary conditions must be de ned for these subdomains. One such arti cal boundaryconditionswhich c...

We analyze self-focusing and singularity formation in the complex Ginzburg-Landau equation (CGL) in the regime where it is close to the critical nonlinear Schrr odinger equation. Using modulation theory Fibich and Papanicolaou, Phys. Lett. A 239:167{173, 1998], we derive a reduced system of ordinary diierential equations that describes self-focusing in CGL. Analysis of the reduced system shows ...

The linear and nonlinear scale spaces are generalized in the complex domain, by combining the di usion equation with the simpli ed Schr odinger equation. A fundamental solution for the linear case is developed. Preliminary analysis of the complex di usion shows that the generalized di usion has properties of both forward and inverse di usion. An important observation, supported theoretically a...

In this paper we present some stability results for Hamiltonian PDE's which have an exact symmetry, and are perturbations of a linear resonant system. We also give an application to the nonlinear Schrr odinger equation constructing innnitely many families of periodical orbits, one of which is orbitally stable, and innnitely many families of \approximatively invariant" 2-dimensional tori, which ...